Exercise 6, § 3

Suppose that K is a field. Let $g$ be a Lie K-algebra, $x \mapsto x_M$ a finite-dimensional representation of $g$, $(e_1, \ldots, e_n)$ a basis of M and $(e_1^*, \ldots, e_n^*)$ the dual basis.

(a) If $f$ is a linear form on $\bigotimes^r M$, then:

(b) Let $\beta$ be a non-degenerate bilinear form on M which is invariant under $g$. Let $(e'_1, \ldots, e'_n)$ be the basis of M such that $\beta(e_i, e_j') = \delta_{ij}$. Let $f$ be an invariant linear form on $\bigotimes^r M$. Deduce from (a) that

is an invariant element of $\bigotimes^r M$ independent of the choice of basis $(e_i)$.

(c) Let U be the enveloping algebra of $g$ and $U^*$ the dual space of U. The adjoint representation of $g$ can be extended to a representation $x \mapsto x_U$ of $g$ on U defined by $x_U u = xu - ux$ for all $u \in U$. Hence $U^*$ has a $g$-module structure. For an element $f$ of $U^*$ to be invariant, it is necessary and sufficient that $f(uv) = f(vu)$ for all $u, v$ in U.

(d) Let $h$ be a finite-dimensional ideal of $g$ and $\gamma$ an invariant bilinear form on $g$ whose restriction to $h$ is non-degenerate. Let $(y_i)_{1 \leq i \leq n}, (y'_j)_{1 \leq j \leq n}$ be two bases of $h$ such that $\gamma(y_i, y'_j) = \delta_{ij}$. Let r be an integer $\geq 1$. Let $f$ be a linear form on U such that $f(uv) = f(vu)$ for all $u, v$ in U. Deduce from (b) and (c) that

is an element of the centre of U independent of the choice of basis $(y_i)$. In particular recover the Casimir elements.

(e) Let $\theta$ be a permutation of $\{1, \ldots, r\}$. Arguing similarly, show that

is an element of the centre of U independent of the choice of basis $(y_i)$.